Abstract
We study the model of continuous chemical reaction networks (CRNs), consisting of reactions such as \(A+B \mathop {\rightarrow }\limits C+D\) that can transform some continuous, nonnegative real-valued quantity (called a concentration) of chemical species A and B into equal concentrations of C and D. Such a reaction can occur from any state in which both reactants A and B are present, i.e., have positive concentration. We modify the model to allow inhibitors, for instance, reaction \(A+B \xrightarrow []{\begin{array}{c} I\\ \bot \end{array}} C+D\) can occur only if the reactants A and B are present and the inhibitor I is absent.
The computational power of non-inhibitory CRNs has been studied. For instance, the reaction \(X_1+X_2 \mathop {\rightarrow }\limits Y\) can be thought to compute the function \(f(x_1,x_2) = \min (x_1,x_2)\). Under an “adversarial” model in which reaction rates can vary arbitrarily over time, it was found that exactly the continuous, piecewise linear functions can be computed, ruling out even simple functions such as \(f(x) = x^2\). In contrast, in this paper we show that inhibitory CRNs can compute any computable function \(f:\mathbb {N}\rightarrow \mathbb {N}\).
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Notes
- 1.
Technically this is using the so-called dual-rail encoding, which represents a single real value x as the difference of two species concentrations \([X^+] - [X^-]\). If one encodes inputs and output directly as nonnegative concentrations, then some discontinuities can occur, but only when some input \(x_i\) goes from 0 to positive.
- 2.
Note that our notation \(A+B \xrightarrow []{\begin{array}{c} I\\ \bot \end{array}} C+D\) puts inhibitors above the reaction arrow where a rate constant would normally be written, but since we consider rate-independent computation, we will have no rate constants. We also note that in gene regulatory networks, typically a species (called transcription factor in that literature) inhibits another species, which is assumed to be produced at some otherwise constant rate by a single reaction, whereas our model is more general in allowing inhibitors of arbitrary reactions (so I could inhibit production of C via one reaction \(A \xrightarrow []{\begin{array}{c} I\\ \bot \end{array}} C\) but not via another reaction \(B \mathop {\rightarrow }\limits C\).).
- 3.
It is customary to define, for each reaction, a rate constant \(k \in \mathbb {R}_{>0}\) specifying a constant multiplier on the mass-action rate (i.e., the product of the reactant concentrations), but as we are studying CRNs whose output is independent of the reaction rates, we leave the rate constants out of the definition.
- 4.
\(\textbf{M}\) does not fully specify \(\mathcal {C}\), since catalysts and inhibitors are not modeled: reactions \(A + B \xrightarrow []{\begin{array}{c} C\\ \bot \end{array}} A + D\) and \(B \mathop {\rightarrow }\limits D\) both correspond to the column vector \((0,-1,0,1)^\top \).
- 5.
Since iCRNs operate on real-valued concentrations, a very similar definition for functions \(f: \mathbb {R}_{\ge 0}\rightarrow \mathbb {R}_{\ge 0}\) makes sense (and was formally defined for non-inhibitory CRNs in [3]); Sect. 4 discusses this issue further. We could also extend the definition to take multiple inputs for a function \(f:\mathbb {N}^d \rightarrow \mathbb {N}\), but since register machines are Turing universal, we could encode multiple input integers via a pairing function into a single integer, so it is no loss of generality to consider single-input functions.
- 6.
- 7.
The long wave seen in the middle is because the reaction \(C_1 + R_\textrm{in} \xrightarrow []{\begin{array}{c} B_1\\ \bot \end{array}} A_2\), when \(R_\textrm{in}\) starts at 1, has a much slower rate of convergence (linear, compared to exponential convergence when \(R_\textrm{in}\) starts 2 or higher). Consequently, \(C_1\) from time \(\approx 300\) to time \(\approx 800\), despite being “close” to 0, is decaying to 0 much more slowly than in previous oscillations. Thus \(C_1\) much more strongly inhibits the reaction \(A_2 \xrightarrow []{\begin{array}{c} C_1\\ \bot \end{array}} B_2\) than in previous oscillations. \(A_2\) and \(B_2\) are the two species “swapping” very slowly between time 300 and 900.
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Acknowledgements
DD and KC were supported by NSF awards 2211793, 1900931, 1844976, and DoE EXPRESS award SC0024467.
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Calabrese, K., Doty, D. (2024). Rate-Independent Continuous Inhibitory Chemical Reaction Networks Are Turing-Universal. In: Cho, DJ., Kim, J. (eds) Unconventional Computation and Natural Computation. UCNC 2024. Lecture Notes in Computer Science, vol 14776. Springer, Cham. https://doi.org/10.1007/978-3-031-63742-1_8
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